Peel carrots and onions and cut into cubes. Wash sage and thyme, shake dry, pluck leaves from the stalks except for some thyme for garnishing. Peel 2 garlic cloves and cut into fine cubes.
Wash the legs, pat dry and season with salt, pepper, garlic and thyme. Heat 2 tablespoons of oil in a frying pan and fry the legs in portions all around. Remove the legs and sauté the prepared vegetables in the frying fat.
Deglaze with wine and stock. Put legs and sage into the roaster and bring to the boil. Braise in the preheated oven (electric cooker: 150 °C/ convection oven: 125 °C/ gas: stage 1) for 2-2 1/2 hours. For the aioli, peel the remaining garlic.
Puree the garlic, egg, lemon juice and 1/4 teaspoon salt with a hand blender to a homogenous mass. Add 125 ml oil drop by drop while stirring. Peel and slice the potatoes. Heat 250 ml oil and deep-fry the potatoes in portions.
Remove the finished potatoes, drain the oil well and season the potato slices with salt. Remove the rabbit from the sauce. Pour the sauce through a sieve. Arrange rabbit, sauce, potatoes, aioli and garnish with thyme.
Tomato salad tastes good with it.